Problem: Divide the following complex numbers. $ \dfrac{15-15i}{5-5i}$
Answer: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${5+5i}$ $ \dfrac{15-15i}{5-5i} = \dfrac{15-15i}{5-5i} \cdot \dfrac{{5+5i}}{{5+5i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(15-15i) \cdot (5+5i)} {(5-5i) \cdot (5+5i)} = \dfrac{(15-15i) \cdot (5+5i)} {5^2 - (-5i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(15-15i) \cdot (5+5i)} {(5)^2 - (-5i)^2} = $ $ \dfrac{(15-15i) \cdot (5+5i)} {25 + 25} = $ $ \dfrac{(15-15i) \cdot (5+5i)} {50} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({15-15i}) \cdot ({5+5i})} {50} = $ $ \dfrac{{15} \cdot {5} + {-15} \cdot {5 i} + {15} \cdot {5 i} + {-15} \cdot {5 i^2}} {50} $ Evaluate each product of two numbers. $ \dfrac{75 - 75i + 75i - 75 i^2} {50} $ Finally, simplify the fraction. $ \dfrac{75 - 75i + 75i + 75} {50} = \dfrac{150 + 0i} {50} = 3 $